Question: Integrate. $\int\left(-\dfrac3x+3e^x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $-3\ln|x|+3+C$ (Choice B) B $-3\ln|x|+3e^x+C$ (Choice C) C $\ln|-3|+3+C$ (Choice D) D $\ln|-3|+3e^x+C$
Solution: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(-\dfrac3x+3e^x \right)dx \\\\ &=-3\int \dfrac1x\,dx+3\int e^x \,dx \\\\ &=-3\ln|x|+3e^x+C \end{aligned}$